\begin{frame}
\frametitle{Extended XMG with control \cite{krithivasan1977variations}}

\begin{define}[XMG with control \cite{krithivasan1977variations}]
Let $G = (G_H, G_V)$ be an Ex-XMG ($X \in \{PS, CS, CF, R\}$) with intermediates $I = \{S_1, \dots, S_n\}$

Let $I' = \{M_1, \dots, M_n\}$ with each $M_i$ corresponds to a $S_i$. We define: 
\begin{itemize}
	\item a control word $w \in I'^*$
	\item a control set $C \subseteq I'^*$
\end{itemize}

Being a control word means, that the first vertical derivation is done for the first $M_{i_1}$ in w. Next, the derivation for $M_{i_2}$ is done and so on. 

We denote
\begin{itemize}
	\item $M_w(G)$ set of arrays generated over G with control word w. 
	\item $M_C(G)$ set of arrays generated over G with any control word of C. 
\end{itemize}
\end{define}

\end{frame}

\begin{frame}
\frametitle{Example}

\begin{Example}
$M(G) = (L)::(R_1, R_2)$  with attach symbols $(d, d)$. 
\begin{itemize}
	\item $L = \{S_1^nS_2^nS_1^n \vert n \geq 1\}$
	\item $R_1 = .^+$
	\item $R_2 = X^+$
\end{itemize}

and control set $C = \{M_1^nM_2^{2n}\ \vert n \geq 1\}$
\end{Example}

Then M(G) generate T-shaped figures with the vertical line out of X's and the horizontal line out of .'s. 

\end{frame}

\begin{frame}
\frametitle{Example derivation}

\[
S
\overset{*}{\Rightarrow}
\boxed{
\begin{aligned}
\begin{matrix}
S_1 & S_1 & S_2 & S_2 & S_1 & S_1
\end{matrix}
\end{aligned}
}
\]

\[
\overset{*}{\Downarrow}
\boxed{
\begin{aligned}
\begin{matrix}
. & . & S_2 & S_2 & . & . \\[-0.5ex]
. & . &  &  & . & . \\[-0.5ex]
. & . &  &  & . & .
\end{matrix}
\end{aligned}
}
\overset{*}{\Downarrow}
\boxed{
\begin{aligned}
\begin{matrix}
. & . & X & X & . & . \\[-0.5ex]
. & . & X & X & . & . \\[-0.5ex]
. & . & X & X & . & . \\[-0.5ex]
 &  & X & X &  &  \\[-0.5ex]
 &  & X & X &  &  \\[-0.5ex]
 &  & X & X &  & 
\end{matrix}
\end{aligned}
}
\]

\end{frame}

\begin{frame}
\frametitle{Hierarchy}

Let $\mathcal{L}(i, j)$ the family of extended matrix models of type i with control of type j ($i, j \in \{0, 1, 2, 3\}$). 

\begin{thm}
\begin{itemize}
	\item $\mathcal{L}(i + 1, j) \subsetneq \mathcal{L}(i, j)$ for $i \in \{0, 1, 2\}, j \in \{0, 1, 2, 3\}$
	\item $\mathcal{L}(i, 3) = \mathcal{L}(i, 2)$ for $i \in \{0, 1, 2, 3\}$
	\item $\mathcal{L}(i, 2) \subsetneq \mathcal{L}(i, 1)$, for $i \in \{1, 2, 3\}$
\end{itemize}
	
\end{thm}

\begin{proof}
see \cite{krithivasan1977variations}
\end{proof}

\end{frame}

\begin{frame}[allowframebreaks]
\frametitle{Variation of extended matrix grammars}

Variation 1: 
\begin{itemize}
	\item Start the upward derivation of a picture one line above the line, where downwards derivation starts. 
	\item We can also use control on this model. 
\end{itemize}

\ \\

Variation 2:
\begin{itemize}
	\item Use variation 1 and extend this in that way, that each intermediate can also have two vertical derivations. 
\end{itemize}

\pagebreak

\begin{columns}
\begin{column}[l]{8cm}
Variation 3: 
\begin{itemize}
	\item Use variation 2 and generalize the use of the attached symbols. Therefore we allow any intermediates to have more than two attached symbols. We do not distinguish up and down anymore but each intermediate is attached with a set of different angles, in which directions the corresponding derivations are starting. With this variation, 3-dimensional objects can be created. 
\end{itemize}
\end{column}
\begin{column}[r]{2cm}
\includegraphics[scale=0.2]{img/3d_generated_object.png}
\end{column}
\end{columns}

\end{frame}